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The geometry of quasicrystals. (English. Russian original) Zbl 0807.52017

Russ. Math. Surv. 48, No. 1, 37-100 (1993); translation from Usp. Mat. Nauk. 48, No. 1(289), 41-102 (1993).
Some geometric properties of quasicrystals are summarized. Past I of the article focusses on group theoretical aspects and part II on matching rules.
The exposition is slightly out of date as various developments are not mentioned which are published in physics journals. In particular, the entire discussion about the existence of local matching rules suffers from not using the equivalence concept of mutual local derivability [cf. the reviewer, M. Schlottmann and P. D. Jarvis, J. Phys. A, Math. Gen. 24, No. 19, 4637-4654 (1991; Zbl 0755.52006)]. Also, the section method (chapter 2.3) was developed by P. Kramer [Mod. Phys. Lett. B 1, No. 1-2, 7-18 (1987) and J. Math. Phys. 29, No. 2, 516-525 (1988; Zbl 0655.52011)] and has found a much more general and rigorous formulation than stated in this article [cf. P. Kramer and M. Schlottmann, J. Phys. A, Math. Gen. 22, No. 23, L1097–L1102 (1989; Zbl 0719.52011) and M. Schlottmann, Int. J. Mod. Phys. B 7, No. 6-7, 1351-1363 (1993; Zbl 0798.52030)].
For expositions of more recent developments, the reader is refered to the reviewer’s volumes “Selected Topics in the Theory of Quasicrystals”, ed. M. Baake, published as special issue of Int. J. Mod. Phys. B 7 (1993) and “Quasicrystals – The State of the Art”, eds. D. P. Di Vincenzo and P.J. Steinhardt, World Scientific, Singapore (1991).

MSC:

52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
82D25 Statistical mechanics of crystals
20F99 Special aspects of infinite or finite groups
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